**Instructor:** Victor Guillemin

**Office:** Room 2-270

**Email:** vwg [at] math . mit .edu

**Lectures:** M W F, 11am, 2-151

**Office Hours:** M W F, 2pm - 3pm

## COURSE DESCRIPTION

Multilinear algebra: tensors and exterior forms. Differential forms on R^n: exterior differentiation, the pull-back operation and the Poincaré lemma. Applications to physics: Maxwell’s equations from the differential form perspective. Integration of forms on open sets of R^n. The change of variables formula revisited. The degree of a differentiable mapping. Differential forms on manifolds and De Rham theory. Integration of forms on manifolds and Stokes’ theorem. The push-forward operation for forms. Thom forms and intersection theory. Applications to differential topology.

**Prerequisites:** 18.101; 18.700 or 18.701

**Text Book:** The primary text for the course will be the notes prepared by Professor Guillemin and posted here on the web. Some useful secondary references include Spivak's *Calculus on Manifolds*, Munkres's *Analysis on Manifolds*, and Guillemin and Pollack's *Differential Topology*.